In the following we set \(\delta \)  to be a small constant for convenience. We find that the energy spectrum of the excited band is complex and the energy spectrum of ground band is real: \(E_+ = d(\vec{k})-2i\Gamma (\vec{k})\)  and \(E_- = -d(\vec{k})\) , where \(\Gamma (\vec{k})=\sqrt{\sum _{i=x,y,z} \Gamma _i^2(\vec{k})}\) . The lowest real part of the eigenspectrum gives the effective ground band, and the imaginary part of energy is always non-positive that is the decay rate of the excited eigenstate [1]}, [2]}, [3]}, [4]}. It is clear that the decay rate \(\gamma (\vec{k})\)  depends on the Bloch-vector \(\vec{k}\) , which is different from the previous studies. This result suggests that the decay rate of the upper Bloch band is different at different positions in momentum space, leading to interesting features in the Hall conductance.
